Impurities in systems of noninteracting trapped fermions

Dean, David; Le Doussal, Pierre; Majumdar, Satya; Schehr, Grégory

doi:10.21468/SciPostPhys.10.4.082

SciPost Physics

Impurities in systems of noninteracting trapped fermions

David S. Dean, Pierre Le Doussal, Satya N. Majumdar, Gregory Schehr

SciPost Phys. 10, 082 (2021) · published 14 April 2021

doi: 10.21468/SciPostPhys.10.4.082
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Abstract

We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-P\'ech\'e (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.10.4.082
TI  - Impurities in systems of noninteracting trapped fermions
PY  - 2021/04/14
UR  - https://scipost.org/SciPostPhys.10.4.082
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 10
IS  - 4
SP  - 082
A1  - Dean, David
AU  - Le Doussal, Pierre
AU  - Majumdar, Satya
AU  - Schehr, Grégory
AB  - We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-P\'ech\'e (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.
ER  -

@Article{10.21468/SciPostPhys.10.4.082,
	title={{Impurities in systems of noninteracting trapped fermions}},
	author={David S. Dean and Pierre Le Doussal and Satya N. Majumdar and Gregory Schehr},
	journal={SciPost Phys.},
	volume={10},
	pages={082},
	year={2021},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.10.4.082},
	url={https://scipost.org/10.21468/SciPostPhys.10.4.082},
}

Cited by 7

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¹ David Dean,
² Pierre Le Doussal,
³ Satya Majumdar,
³ Grégory Schehr

Funders for the research work leading to this publication